Two Variables
With two unknowns you need two independent equations to pin down a unique pair (x, y). The two everyday tools are substitution and elimination. Substitution works best when one variable already has a coefficient of 1: solve that equation for it and plug into the other. Elimination shines when coefficients line up: scale the equations so one variable has equal and opposite coefficients, then add. A single equation in two variables, like 2x + 3y = 12, has infinitely many solutions — it is a line, not a point — so CAT sometimes hands you one equation and an extra hidden constraint (positive integers, a ratio, a known sum) to force a unique answer. When the question asks only for an expression such as x + y or x − y, do not solve fully; add or subtract the two equations to get the combination directly. That combination shortcut saves a surprising amount of time in the exam.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Single and standard forms
| Linear equation (one variable) | ax + b = 0 ⇒ x = −b/a (a ≠ 0) |
|---|---|
| Two-variable standard form | a₁x + b₁y = c₁ and a₂x + b₂y = c₂ |
| Slope–intercept line | y = mx + c (m = slope, c = y-intercept) |
| Cross-multiplication solution | x = (b₁c₂ − b₂c₁)/(a₁b₂ − a₂b₁) |
| Companion value | y = (c₁a₂ − c₂a₁)/(a₁b₂ − a₂b₁) |
Solution conditions for a pair
| Unique solution (lines meet) | a₁/a₂ ≠ b₁/b₂ |
|---|---|
| No solution (parallel lines) | a₁/a₂ = b₁/b₂ ≠ c₁/c₂ |
| Infinite solutions (same line) | a₁/a₂ = b₁/b₂ = c₁/c₂ |
| Determinant test | D = a₁b₂ − a₂b₁; D ≠ 0 ⇒ unique |
| n equations need | n independent equations for n unknowns |